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The differential quadrature solution of nonlinear reaction–diffusion and wave equations using several time‐integration schemes
Author(s) -
Meral Gülnihal,
TezerSezgin M.
Publication year - 2011
Publication title -
international journal for numerical methods in biomedical engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.741
H-Index - 63
eISSN - 2040-7947
pISSN - 2040-7939
DOI - 10.1002/cnm.1305
Subject(s) - finite element method , discretization , quadrature (astronomy) , nyström method , nonlinear system , finite difference method , mathematics , finite difference , mathematical analysis , partial differential equation , boundary value problem , relaxation (psychology) , numerical integration , physics , psychology , social psychology , quantum mechanics , optics , thermodynamics
Three different time‐integration schemes, namely the finite difference method (FDM) with a relaxation parameter, the least‐squares method (LSM) and the finite element method (FEM), are applied to the differential quadrature (DQM) solution of one‐dimensional nonlinear reaction–diffusion and wave equations. In the solution procedure, the space derivatives are discretized using DQM, which may also be used without the need of boundary conditions. The aim of the paper is to find computationally more efficient time‐integration algorithm by comparison. For the nonlinear reaction–diffusion equation, which is a parabolic type, FEM is found to be the method of choice in terms of accuracy and computational cost. For the nonlinear wave equation, it is seen that both the FDM and LSM are better than FEM. Therefore, LSM can be preferred as a direct method for the nonlinear wave equation. Copyright © 2009 John Wiley & Sons, Ltd.